New approach to the numerical solution of forward-backward equations

Abstract

This paper is concerned with the approximate solution of functional differential equations having the form:which takes given values on intervals [−1, 0] and (k − 1, k]. We introduce and analyse some new computational methods for the solution of this problem. Numerical results are presented and compared with the results obtained by other methods.

An abstract framework in the numerical solution of boundary value problems for neutral functional differential equations

“…By (53), (48), Remark 4 and Theorem 3, we obtain that there exists a positive integer K 2 such that, for any positive integer K ≥ K 2 , D * Ψ K is invertible and…”

“…BVPs for non-neutral delay differential equations [8,12,28,29,37,38,49,52] BVPs for non-neutral differential equations with deviating arguments [1,3,[9][10][11]19,20,50,51,53] BVPs for non-neutral differential equations with a state-dependent deviating argument θ(t, y(t)) [4] BVPs for non-neutral singularly perturbed differential equations with deviating arguments [31][32][33][34][35] BVPs for the determination of periodic solutions of non-neutral delay differential equations [15,16,18,39,54] BVPs for the determination of periodic solutions of non-neutral delay differential equations with a state-dependent delay [40] BVPs for the determination of periodic solutions of non-neutral differential equations with deviating arguments [6,7] BVPs for non-neutral Fredholm integro-differential equations [21,22,45] BVPs for non-neutral Fredholm integro-differential equations with weakly singular kernels [46][47][48] BVPs for non-neutral Volterra integro-differential equations [23] BVPs for the determination of periodic solutions of neutral delay differential equations [5,17] BVPs for neutral differential equations with st...…”

Finite element solution of a linear mixed-type functional differential equation

“…Three alternative methods were used for this purpose. The collocation (COL) and least squares (LS) methods are described in [11,12,18,19]. Here we will describe in detail the finite element (FE) method.…”

“…Assuming that such a solution exists they have introduced a numerical algorithm to compute it. In [18], a numerical algorithm based on the collocation method was proposed for the solution of such BVPs. In [11,12,19,20] these methods were extended to the non-autonomous case and a new algorithm, based on the least squares method, was introduced.…”

“…Here we provide its detailed description and error analysis. A comparative analysis of the algorithms presented in [7,11,12,[18][19][20] is also provided. We introduce a criterion for the existence of a solution of the considered boundary value problem and present some examples that illustrate the application of this criterion.…”